Since the model (REF ) only contains nearest neighbor interactions, naturally only nearest neighbor coupling terms need to be considered in the mean field Hamiltonian. Suppose that the QSL contains zero fluxes in each hexagon (it has been shown that the flux states are not energetically favored[1]}), and that \(C_3\)  symmetry is preserved in the mean field Hamiltonian. The most general mean field Hamiltonian take the following form,
\( H_{\rm mf}^{\rm SL}\!\!\!&=&\!\!\!\!\!\!\!\!\sum _{\langle ij \rangle \in \alpha \beta (\gamma )}\!\!\Big \lbrace [f_i^\dagger (t_1^\gamma R_{\alpha \beta } + t_0^\gamma + t_2^\gamma \sigma ^\gamma + t_3^\gamma \sigma ^\gamma R_{\alpha \beta })f_j \nonumber \\&&\!\!\! + f_i^\dagger (\Delta _1^\gamma R_{\alpha \beta } + \Delta _0^\gamma +\Delta _2^\gamma \sigma ^\gamma + \Delta _3^\gamma \sigma ^\gamma R_{\alpha \beta })\bar{f}_j] + {\rm H.c.}\Big \rbrace  \nonumber \\&&\!\!\! + \lambda \sum _i f_i^\dag f_i + {\rm const},\) 
